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Properties

Label 2-370-185.88-c1-0-1
Degree $2$
Conductor $370$
Sign $0.505 - 0.862i$
Analytic cond. $2.95446$
Root an. cond. $1.71885$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (−2.78 − 0.745i)3-s + (−0.499 − 0.866i)4-s + (−2.02 − 0.947i)5-s + (2.03 − 2.03i)6-s + (−4.90 − 1.31i)7-s + 0.999·8-s + (4.58 + 2.64i)9-s + (1.83 − 1.27i)10-s − 0.446i·11-s + (0.745 + 2.78i)12-s + (2.31 + 4.01i)13-s + (3.59 − 3.59i)14-s + (4.92 + 4.14i)15-s + (−0.5 + 0.866i)16-s + (−0.535 − 0.309i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−1.60 − 0.430i)3-s + (−0.249 − 0.433i)4-s + (−0.905 − 0.423i)5-s + (0.831 − 0.831i)6-s + (−1.85 − 0.496i)7-s + 0.353·8-s + (1.52 + 0.881i)9-s + (0.579 − 0.404i)10-s − 0.134i·11-s + (0.215 + 0.802i)12-s + (0.642 + 1.11i)13-s + (0.960 − 0.960i)14-s + (1.27 + 1.07i)15-s + (−0.125 + 0.216i)16-s + (−0.129 − 0.0750i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.505 - 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.505 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.505 - 0.862i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (273, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.505 - 0.862i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.228667 + 0.131010i\)
\(L(\frac12)\) \(\approx\) \(0.228667 + 0.131010i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (2.02 + 0.947i)T \)
37 \( 1 + (3.72 + 4.81i)T \)
good3 \( 1 + (2.78 + 0.745i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (4.90 + 1.31i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + 0.446iT - 11T^{2} \)
13 \( 1 + (-2.31 - 4.01i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.535 + 0.309i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.21 + 4.52i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 - 2.27T + 23T^{2} \)
29 \( 1 + (4.05 - 4.05i)T - 29iT^{2} \)
31 \( 1 + (-0.176 - 0.176i)T + 31iT^{2} \)
41 \( 1 + (3.38 - 1.95i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 - 8.15T + 43T^{2} \)
47 \( 1 + (6.46 - 6.46i)T - 47iT^{2} \)
53 \( 1 + (-8.08 + 2.16i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (3.85 - 1.03i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-2.05 + 7.66i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-3.53 + 13.2i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-7.60 - 13.1i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.847 + 0.847i)T - 73iT^{2} \)
79 \( 1 + (2.81 - 10.5i)T + (-68.4 - 39.5i)T^{2} \)
83 \( 1 + (9.97 - 2.67i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-0.257 - 0.960i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 - 3.22iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.36588957920970682951233448021, −10.89232739919706234801134015894, −9.650233205139512017991886682313, −8.857416720355165910415439994279, −7.23888594688845899256709648699, −6.86489881673342303583913850523, −6.04673857130525290415606562882, −4.89993984058995289365619670062, −3.71556467336781568204849943425, −0.791404559749989246091046223945, 0.38871130758445867060161597472, 3.15189303323868378849466160965, 3.97626241794033051173882642839, 5.52630686951255242223111354008, 6.30755410148429239269587258023, 7.30652849377923906279128532896, 8.681050185687633021781422737754, 9.970878597214005992486663269962, 10.30000399581025475158167063856, 11.23930304813968545867014608822

Graph of the $Z$-function along the critical line